RE: [tuning] Re: Towards a hyper MOS and Dissertation on MOS, Ste inhaus, etc

Dave Keenan wrote,

>It seems that the denominators of the convergents give the
>cardinalities of the strictly proper scales, while the
>semi-convergents give improper (or merely proper) scales. Do you know
>whether that is always the case?

I stated as much in this summer's posts here in reply to Jason Yust, but I
don't know if I had a mental proof or not at the time. My mind is far too
fuzzy now.

Thanks for this spreadsheet Dave. I don't know how I feel about your minimax
beat rate criterion -- your meantone fifth of 695.63 cents is smaller than
the Zarlino (2/7-comma meantone) fifth -- besides the other objections we've
raised to using beat rate without any provision for beat amplitude, it may
be that you're focusing too narrowly on only one voicing of one chord (the
4:5:6), leading to such a skewed result.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> Thanks for this spreadsheet Dave. I don't know how I feel about your
minimax
> beat rate criterion -- your meantone fifth of 695.63 cents is
smaller than
> the Zarlino (2/7-comma meantone) fifth -- besides the other
objections we've
> raised to using beat rate without any provision for beat amplitude,
it may
> be that you're focusing too narrowly on only one voicing of one
chord (the
> 4:5:6), leading to such a skewed result.

My approach has been to maximise the consonance of the most consonant
chord. But you're right. The 695.63 meantone fifth shows clearly that
minimax beat rate is not doing that. It is too hard on the fifths. I
calculated those optima ages ago.

This is getting off the topic of this thread and onto something
JdL was chasing. What would be a simple way to include some sort of
"typical" beat amplitudes in the optimisation? I feel that the
tolerance for mistuning is greatest for intervals with a*b complexity
of about 30 to 35 and less for both simpler and more complex ratios.

Note that the weighting of the cents values implied by beat-rate alone
is (to a good approximation for deviations of less than 20 cents) the
LCM of the interval, as it appears in the extended ratio of the chord
(not reduced to lowest terms). If the interval appears more than once
e.g. 4:6 and 6:9 in 4:5:6:7:9, then the highest LCM is used, i.e.
LCM(6,9) = 18.

Regards,
-- Dave Keenan

Dave, I think tolerance for dyads within larger chords will turn out to be a
far more complex issue than just weighting beat rates (consider the large
differences between otonal/utonal partners, for example). For single
intervals, I'm not sure how beat rates will help you with this:

>I feel that the
>tolerance for mistuning is greatest for intervals with a*b complexity
>of about 30 to 35 and less for both simpler and more complex ratios.

though I thought we had once discussed using a minimum of two functions for
the tolerance (to get it to behave this way), one being based on beat rate
and another on . . . ?